Question

In how many ways can the letters in the word "Monday" be arranged?

Short answer

There are 720 different ways that the letters in the word "Monday" can be arranged.

Step-by-step solution

Understand the permutation formula

In a permutation, the order of items matters. The formula to calculate the number of permutations of a set of items is \( n! \), where \( n \) is the total number of items, and \( ! \) indicates a factorial operation. A factorial operation multiplies the number by every integer less than it down to 1. For example, \( 5! \) is \( 5*4*3*2*1 = 120 \).

Apply the permutation formula to the problem

"Monday" has 6 letters, meaning we have \( n = 6 \). Each letter can take any position, and each position can be filled by any letter. Therefore, according to the permutation formula, there are \( 6! \) ways to arrange the letters in the word 'Monday'.

Calculate the number of permutations

Using a calculator (or manually if necessary), calculate \( 6! \) = \( 6*5*4*3*2*1 = 720 \).

Present the final answer

So, there are 720 different ways that the letters in the word "Monday" can be arranged.

Key concepts

Factorial

The concept of a factorial is a fundamental part of mathematics, especially in permutations and combinations. The factorial of any non-negative integer \( n \) is the product of all the positive integers less than or equal to \( n \). This is represented as \( n! \).
For example:

• \( 4! = 4 \times 3 \times 2 \times 1 = 24 \)
• \( 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \)

Interestingly, \( 0! \), the factorial of zero, is defined as 1. This might seem strange initially, but it is crucial to making mathematical formulas work correctly.
Understanding factorials is essential for solving problems related to permutations, where you calculate how items can be arranged in various orders.

Arrangements

In mathematics, understanding arrangements involves knowing how different items can be ordered or placed. When dealing with the word "Monday," arrangements refer to changing the order of its letters.
For six distinct letters, like in "Monday," the number of possible arrangements or orderings is determined using the permutation formula. Since the sequence of items matters in permutations, all possible unique sequences are counted.

• Each position in the arrangement can be filled by any letter from the set of available letters.
• The total arrangement possibilities are calculated by determining the factorial of the number of letters.

Using "Monday" as an example, we calculate \(6!\) because there are 6 letters, resulting in 720 unique arrangements.

Combinatorics

Combinatorics is the branch of mathematics focused on counting and arranging objects. It includes topics like permutations, combinations, and factorials, which all rely on counting principles.
In the context of permutations, combinatorics helps determine how many different orders items can be placed in. For the word "Monday," combinatorics is used to calculate the number of ways its 6 letters can be rearranged.

• Permutations consider the order of items, meaning different sequences count as different permutations.
• Combinatorics principles use factorials to compute these arrangements, as seen in the solution of finding \(6!\).

By studying combinatorics, one learns how to apply these counting techniques to various problems, enhancing understanding and problem-solving skills in mathematics.